48 1. Random Walk and Discrete Heat Equation

Prove the following estimates. The constants c1,c2,c3 are positive

constants independent of N and the estimates should hold for all N

and all j = 1,...,N − 1.

•

β(1,N) β(2,N) · · · β(N − 1,N) ≤ N

cosh−1(2).

• There is a c1 such that

cosh

jπ

N

+ cos

jπ

N

− 2 ≤

c1 j4

N 4

.

• There is a c2 such that

|β(j, N) − πj| ≤

c2

j4

N 3

.

• There is a c3 such that

β(j, N) ≥ c3 j.